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The 30 pages that link to Prime-counting function, each with the reason it gives.

• Riemann hypothesisRelated: The hypothesis predicts a strong bound on this function’s deviation from its main estimate.

• Prime gapsRelated: Its local changes reveal how many primes, and thus gaps, occur in an interval.

• Twin prime conjectureRelated: Twin-prime predictions estimate pair counts relative to the overall distribution of primes.

• Goldbach's conjectureRelated: It helps quantify how many candidate prime pairs may represent a given even number.

• Explicit formulaRelated: Explicit formulas describe its deviations from smooth approximations through zero contributions.

• Möbius functionRelated: Explicit formulas connect prime counts with weighted sums of the Möbius function.

• Sieve of EratosthenesRelated: Counting the sieve’s survivors directly gives π(x) for a chosen limit.

• Prime number theorem for arithmetic progressionsRelated: The theorem estimates its residue-class analogue, π(x;q,a), by Li(x)/φ(q).

• Bombieri–Vinogradov theoremRelated: Progression counts are compared with the expected prime-counting main term.

• Integer sequenceRelated: Its values form an integer sequence recording the distribution of primes.

• Atle SelbergRelated: The prime number theorem, whose elementary proof Selberg helped establish, describes its asymptotic growth.

• Chebotarev density theoremRelated: Density is expressed through the asymptotic growth of primes with the selected Frobenius class.

• Legendre's conjectureRelated: Its values express the claim as π((n+1)²) − π(n²) ≥ 1.

• Twin primeRelated: Twin-prime counts can be compared with the overall growth in the number of primes.

• Wilson's theoremRelated: Wilson's theorem can encode primality in formulas, though it offers little computational advantage for counting.

• Siegel–Walfisz theoremRelated: Progression-specific counting functions split this global count among residue classes.

• Brun–Titchmarsh theoremRelated: The theorem bounds a progression-restricted counterpart of this count.

• Bonse's inequalityRelated: It describes how many factors enter the primorial as its endpoint grows.

• Riemann–Siegel formulaRelated: Zeta values computed by the formula inform explicit estimates for prime-counting errors.

• Brocard's conjectureRelated: Subtracting its values at the squared endpoints expresses the prime count in the conjectured interval.

• Divergence of the sum of the reciprocals of the primesRelated: Partial summation connects estimates for prime counts to the growth of reciprocal-prime sums.

• Goldbach–Euler theoremRelated: Understanding prime density informs estimates for how many Goldbach representations should exist.

• Grimm's conjectureRelated: Prime distribution estimates can help bound available divisors in intervals.

• Lemoine's conjectureRelated: Counting candidate primes helps estimate how many decompositions an integer may have.

• Wolstenholme's theoremRelated: Counting exceptional primes asks how their number grows among all primes.